Theorem th3q 5376

Description: Theorem 3Q of [Enderton] p. 60, extended to operations on ordered pairs.

Hypotheses

Ref	Expression
th3q.1	|- R e. _V
th3q.2	|- Er R
th3q.3	|- dom R = (S X. S)
th3q.4	|- ((((w e. S /\ v e. S) /\ (u e. S /\ t e. S)) /\ ((s e. S /\ f e. S) /\ (g e. S /\ h e. S))) -> ((<.w, v>.R<.u, t>. /\ <.s, f>.R<.g, h>.) -> (<.w, v>.F<.s, f>.)R(<.u, t>.F<.g, h>.)))
th3q.5	|- G = {<.<.x, y>., z>. | ((x e. ((S X. S)/.R) /\ y e. ((S X. S)/.R)) /\ E.wE.vE.uE.t((x = [<.w, v>.]R /\ y = [<.u, t>.]R) /\ z = [(<.w, v>.F<.u, t>.)]R))}

Assertion

Ref	Expression
th3q	|- (((A e. S /\ B e. S) /\ (C e. S /\ D e. S)) -> ([<.A, B>.]RG[<.C, D>.]R) = [(<.A, B>.F<.C, D>.)]R)

Distinct variable groups:   x,y,z,w,v,u,t,s,f,g,h,R   x,S,y,z,w,v,u,t,s,f,g,h   x,A,y,z,w,v,u,t,s,f,g,h   x,B,y,z,w,v,u,t,s,f,g,h   x,C,y,z,w,v,u,t,s,f,g,h   x,D,y,z,w,v,u,t,s,f,g,h   x,F,y,z,w,v,u,t,s,f,g,h   x,G,y,z,w,v,u,t,s,f,g,h

Proof of Theorem th3q

Step	Hyp	Ref	Expression
1		opelxpi 4040	. . . 4 |- ((A e. S /\ B e. S) -> <.A, B>. e. (S X. S))
2		th3q.1	. . . . 5 |- R e. _V
3	2	ecelqsi 5350	. . . 4 |- (<.A, B>. e. (S X. S) -> [<.A, B>.]R e. ((S X. S)/.R))
4	1, 3	syl 12	. . 3 |- ((A e. S /\ B e. S) -> [<.A, B>.]R e. ((S X. S)/.R))
5		opelxpi 4040	. . . 4 |- ((C e. S /\ D e. S) -> <.C, D>. e. (S X. S))
6	2	ecelqsi 5350	. . . 4 |- (<.C, D>. e. (S X. S) -> [<.C, D>.]R e. ((S X. S)/.R))
7	5, 6	syl 12	. . 3 |- ((C e. S /\ D e. S) -> [<.C, D>.]R e. ((S X. S)/.R))
8	4, 7	anim12i 360	. 2 |- (((A e. S /\ B e. S) /\ (C e. S /\ D e. S)) -> ([<.A, B>.]R e. ((S X. S)/.R) /\ [<.C, D>.]R e. ((S X. S)/.R)))
9		eqid 1884	. . . 4 |- [<.A, B>.]R = [<.A, B>.]R
10		eqid 1884	. . . 4 |- [<.C, D>.]R = [<.C, D>.]R
11	9, 10	pm3.2i 307	. . 3 |- ([<.A, B>.]R = [<.A, B>.]R /\ [<.C, D>.]R = [<.C, D>.]R)
12		eqid 1884	. . 3 |- [(<.A, B>.F<.C, D>.)]R = [(<.A, B>.F<.C, D>.)]R
13		opeq12 3160	. . . . . 6 |- ((w = A /\ v = B) -> <.w, v>. = <.A, B>.)
14		eceq2 5336	. . . . . . . . 9 |- (<.w, v>. = <.A, B>. -> [<.w, v>.]R = [<.A, B>.]R)
15	14	eqeq2d 1895	. . . . . . . 8 |- (<.w, v>. = <.A, B>. -> ([<.A, B>.]R = [<.w, v>.]R <-> [<.A, B>.]R = [<.A, B>.]R))
16	15	anbi1d 679	. . . . . . 7 |- (<.w, v>. = <.A, B>. -> (([<.A, B>.]R = [<.w, v>.]R /\ [<.C, D>.]R = [<.C, D>.]R) <-> ([<.A, B>.]R = [<.A, B>.]R /\ [<.C, D>.]R = [<.C, D>.]R)))
17		opreq1 4889	. . . . . . . . 9 |- (<.w, v>. = <.A, B>. -> (<.w, v>.F<.C, D>.) = (<.A, B>.F<.C, D>.))
18		eceq2 5336	. . . . . . . . 9 |- ((<.w, v>.F<.C, D>.) = (<.A, B>.F<.C, D>.) -> [(<.w, v>.F<.C, D>.)]R = [(<.A, B>.F<.C, D>.)]R)
19	17, 18	syl 12	. . . . . . . 8 |- (<.w, v>. = <.A, B>. -> [(<.w, v>.F<.C, D>.)]R = [(<.A, B>.F<.C, D>.)]R)
20	19	eqeq2d 1895	. . . . . . 7 |- (<.w, v>. = <.A, B>. -> ([(<.A, B>.F<.C, D>.)]R = [(<.w, v>.F<.C, D>.)]R <-> [(<.A, B>.F<.C, D>.)]R = [(<.A, B>.F<.C, D>.)]R))
21	16, 20	anbi12d 690	. . . . . 6 |- (<.w, v>. = <.A, B>. -> ((([<.A, B>.]R = [<.w, v>.]R /\ [<.C, D>.]R = [<.C, D>.]R) /\ [(<.A, B>.F<.C, D>.)]R = [(<.w, v>.F<.C, D>.)]R) <-> (([<.A, B>.]R = [<.A, B>.]R /\ [<.C, D>.]R = [<.C, D>.]R) /\ [(<.A, B>.F<.C, D>.)]R = [(<.A, B>.F<.C, D>.)]R)))
22	13, 21	syl 12	. . . . 5 |- ((w = A /\ v = B) -> ((([<.A, B>.]R = [<.w, v>.]R /\ [<.C, D>.]R = [<.C, D>.]R) /\ [(<.A, B>.F<.C, D>.)]R = [(<.w, v>.F<.C, D>.)]R) <-> (([<.A, B>.]R = [<.A, B>.]R /\ [<.C, D>.]R = [<.C, D>.]R) /\ [(<.A, B>.F<.C, D>.)]R = [(<.A, B>.F<.C, D>.)]R)))
23	22	cla42egv 2366	. . . 4 |- ((A e. S /\ B e. S) -> ((([<.A, B>.]R = [<.A, B>.]R /\ [<.C, D>.]R = [<.C, D>.]R) /\ [(<.A, B>.F<.C, D>.)]R = [(<.A, B>.F<.C, D>.)]R) -> E.wE.v(([<.A, B>.]R = [<.w, v>.]R /\ [<.C, D>.]R = [<.C, D>.]R) /\ [(<.A, B>.F<.C, D>.)]R = [(<.w, v>.F<.C, D>.)]R)))
24		opeq12 3160	. . . . . . 7 |- ((u = C /\ t = D) -> <.u, t>. = <.C, D>.)
25		eceq2 5336	. . . . . . . . . 10 |- (<.u, t>. = <.C, D>. -> [<.u, t>.]R = [<.C, D>.]R)
26	25	eqeq2d 1895	. . . . . . . . 9 |- (<.u, t>. = <.C, D>. -> ([<.C, D>.]R = [<.u, t>.]R <-> [<.C, D>.]R = [<.C, D>.]R))
27	26	anbi2d 678	. . . . . . . 8 |- (<.u, t>. = <.C, D>. -> (([<.A, B>.]R = [<.w, v>.]R /\ [<.C, D>.]R = [<.u, t>.]R) <-> ([<.A, B>.]R = [<.w, v>.]R /\ [<.C, D>.]R = [<.C, D>.]R)))
28		opreq2 4890	. . . . . . . . . 10 |- (<.u, t>. = <.C, D>. -> (<.w, v>.F<.u, t>.) = (<.w, v>.F<.C, D>.))
29		eceq2 5336	. . . . . . . . . 10 |- ((<.w, v>.F<.u, t>.) = (<.w, v>.F<.C, D>.) -> [(<.w, v>.F<.u, t>.)]R = [(<.w, v>.F<.C, D>.)]R)
30	28, 29	syl 12	. . . . . . . . 9 |- (<.u, t>. = <.C, D>. -> [(<.w, v>.F<.u, t>.)]R = [(<.w, v>.F<.C, D>.)]R)
31	30	eqeq2d 1895	. . . . . . . 8 |- (<.u, t>. = <.C, D>. -> ([(<.A, B>.F<.C, D>.)]R = [(<.w, v>.F<.u, t>.)]R <-> [(<.A, B>.F<.C, D>.)]R = [(<.w, v>.F<.C, D>.)]R))
32	27, 31	anbi12d 690	. . . . . . 7 |- (<.u, t>. = <.C, D>. -> ((([<.A, B>.]R = [<.w, v>.]R /\ [<.C, D>.]R = [<.u, t>.]R) /\ [(<.A, B>.F<.C, D>.)]R = [(<.w, v>.F<.u, t>.)]R) <-> (([<.A, B>.]R = [<.w, v>.]R /\ [<.C, D>.]R = [<.C, D>.]R) /\ [(<.A, B>.F<.C, D>.)]R = [(<.w, v>.F<.C, D>.)]R)))
33	24, 32	syl 12	. . . . . 6 |- ((u = C /\ t = D) -> ((([<.A, B>.]R = [<.w, v>.]R /\ [<.C, D>.]R = [<.u, t>.]R) /\ [(<.A, B>.F<.C, D>.)]R = [(<.w, v>.F<.u, t>.)]R) <-> (([<.A, B>.]R = [<.w, v>.]R /\ [<.C, D>.]R = [<.C, D>.]R) /\ [(<.A, B>.F<.C, D>.)]R = [(<.w, v>.F<.C, D>.)]R)))
34	33	cla42egv 2366	. . . . 5 |- ((C e. S /\ D e. S) -> ((([<.A, B>.]R = [<.w, v>.]R /\ [<.C, D>.]R = [<.C, D>.]R) /\ [(<.A, B>.F<.C, D>.)]R = [(<.w, v>.F<.C, D>.)]R) -> E.uE.t(([<.A, B>.]R = [<.w, v>.]R /\ [<.C, D>.]R = [<.u, t>.]R) /\ [(<.A, B>.F<.C, D>.)]R = [(<.w, v>.F<.u, t>.)]R)))
35	34	2eximdv 1671	. . . 4 |- ((C e. S /\ D e. S) -> (E.wE.v(([<.A, B>.]R = [<.w, v>.]R /\ [<.C, D>.]R = [<.C, D>.]R) /\ [(<.A, B>.F<.C, D>.)]R = [(<.w, v>.F<.C, D>.)]R) -> E.wE.vE.uE.t(([<.A, B>.]R = [<.w, v>.]R /\ [<.C, D>.]R = [<.u, t>.]R) /\ [(<.A, B>.F<.C, D>.)]R = [(<.w, v>.F<.u, t>.)]R)))
36	23, 35	sylan9 517	. . 3 |- (((A e. S /\ B e. S) /\ (C e. S /\ D e. S)) -> ((([<.A, B>.]R = [<.A, B>.]R /\ [<.C, D>.]R = [<.C, D>.]R) /\ [(<.A, B>.F<.C, D>.)]R = [(<.A, B>.F<.C, D>.)]R) -> E.wE.vE.uE.t(([<.A, B>.]R = [<.w, v>.]R /\ [<.C, D>.]R = [<.u, t>.]R) /\ [(<.A, B>.F<.C, D>.)]R = [(<.w, v>.F<.u, t>.)]R)))
37	11, 12, 36	mp2ani 764	. 2 |- (((A e. S /\ B e. S) /\ (C e. S /\ D e. S)) -> E.wE.vE.uE.t(([<.A, B>.]R = [<.w, v>.]R /\ [<.C, D>.]R = [<.u, t>.]R) /\ [(<.A, B>.F<.C, D>.)]R = [(<.w, v>.F<.u, t>.)]R))
38		ecexg 5322	. . . 4 |- (R e. _V -> [(<.A, B>.F<.C, D>.)]R e. _V)
39	2, 38	ax-mp 7	. . 3 |- [(<.A, B>.F<.C, D>.)]R e. _V
40		eqeq1 1890	. . . . . 6 |- (x = [<.A, B>.]R -> (x = [<.w, v>.]R <-> [<.A, B>.]R = [<.w, v>.]R))
41	40	anbi1d 679	. . . . 5 |- (x = [<.A, B>.]R -> ((x = [<.w, v>.]R /\ y = [<.u, t>.]R) <-> ([<.A, B>.]R = [<.w, v>.]R /\ y = [<.u, t>.]R)))
42	41	anbi1d 679	. . . 4 |- (x = [<.A, B>.]R -> (((x = [<.w, v>.]R /\ y = [<.u, t>.]R) /\ z = [(<.w, v>.F<.u, t>.)]R) <-> (([<.A, B>.]R = [<.w, v>.]R /\ y = [<.u, t>.]R) /\ z = [(<.w, v>.F<.u, t>.)]R)))
43	42	4exbidv 1661	. . 3 |- (x = [<.A, B>.]R -> (E.wE.vE.uE.t((x = [<.w, v>.]R /\ y = [<.u, t>.]R) /\ z = [(<.w, v>.F<.u, t>.)]R) <-> E.wE.vE.uE.t(([<.A, B>.]R = [<.w, v>.]R /\ y = [<.u, t>.]R) /\ z = [(<.w, v>.F<.u, t>.)]R)))
44		eqeq1 1890	. . . . . 6 |- (y = [<.C, D>.]R -> (y = [<.u, t>.]R <-> [<.C, D>.]R = [<.u, t>.]R))
45	44	anbi2d 678	. . . . 5 |- (y = [<.C, D>.]R -> (([<.A, B>.]R = [<.w, v>.]R /\ y = [<.u, t>.]R) <-> ([<.A, B>.]R = [<.w, v>.]R /\ [<.C, D>.]R = [<.u, t>.]R)))
46	45	anbi1d 679	. . . 4 |- (y = [<.C, D>.]R -> ((([<.A, B>.]R = [<.w, v>.]R /\ y = [<.u, t>.]R) /\ z = [(<.w, v>.F<.u, t>.)]R) <-> (([<.A, B>.]R = [<.w, v>.]R /\ [<.C, D>.]R = [<.u, t>.]R) /\ z = [(<.w, v>.F<.u, t>.)]R)))
47	46	4exbidv 1661	. . 3 |- (y = [<.C, D>.]R -> (E.wE.vE.uE.t(([<.A, B>.]R = [<.w, v>.]R /\ y = [<.u, t>.]R) /\ z = [(<.w, v>.F<.u, t>.)]R) <-> E.wE.vE.uE.t(([<.A, B>.]R = [<.w, v>.]R /\ [<.C, D>.]R = [<.u, t>.]R) /\ z = [(<.w, v>.F<.u, t>.)]R)))
48		eqeq1 1890	. . . . 5 |- (z = [(<.A, B>.F<.C, D>.)]R -> (z = [(<.w, v>.F<.u, t>.)]R <-> [(<.A, B>.F<.C, D>.)]R = [(<.w, v>.F<.u, t>.)]R))
49	48	anbi2d 678	. . . 4 |- (z = [(<.A, B>.F<.C, D>.)]R -> ((([<.A, B>.]R = [<.w, v>.]R /\ [<.C, D>.]R = [<.u, t>.]R) /\ z = [(<.w, v>.F<.u, t>.)]R) <-> (([<.A, B>.]R = [<.w, v>.]R /\ [<.C, D>.]R = [<.u, t>.]R) /\ [(<.A, B>.F<.C, D>.)]R = [(<.w, v>.F<.u, t>.)]R)))
50	49	4exbidv 1661	. . 3 |- (z = [(<.A, B>.F<.C, D>.)]R -> (E.wE.vE.uE.t(([<.A, B>.]R = [<.w, v>.]R /\ [<.C, D>.]R = [<.u, t>.]R) /\ z = [(<.w, v>.F<.u, t>.)]R) <-> E.wE.vE.uE.t(([<.A, B>.]R = [<.w, v>.]R /\ [<.C, D>.]R = [<.u, t>.]R) /\ [(<.A, B>.F<.C, D>.)]R = [(<.w, v>.F<.u, t>.)]R)))
51		th3q.2	. . . 4 |- Er R
52		th3q.3	. . . 4 |- dom R = (S X. S)
53		th3q.4	. . . 4 |- ((((w e. S /\ v e. S) /\ (u e. S /\ t e. S)) /\ ((s e. S /\ f e. S) /\ (g e. S /\ h e. S))) -> ((<.w, v>.R<.u, t>. /\ <.s, f>.R<.g, h>.) -> (<.w, v>.F<.s, f>.)R(<.u, t>.F<.g, h>.)))
54	2, 51, 52, 53	th3qlem2 5374	. . 3 |- ((x e. ((S X. S)/.R) /\ y e. ((S X. S)/.R)) -> E*zE.wE.vE.uE.t((x = [<.w, v>.]R /\ y = [<.u, t>.]R) /\ z = [(<.w, v>.F<.u, t>.)]R))
55		th3q.5	. . 3 |- G = {<.<.x, y>., z>. | ((x e. ((S X. S)/.R) /\ y e. ((S X. S)/.R)) /\ E.wE.vE.uE.t((x = [<.w, v>.]R /\ y = [<.u, t>.]R) /\ z = [(<.w, v>.F<.u, t>.)]R))}
56	39, 43, 47, 50, 54, 55	oprabvali 4954	. 2 |- (([<.A, B>.]R e. ((S X. S)/.R) /\ [<.C, D>.]R e. ((S X. S)/.R)) -> (E.wE.vE.uE.t(([<.A, B>.]R = [<.w, v>.]R /\ [<.C, D>.]R = [<.u, t>.]R) /\ [(<.A, B>.F<.C, D>.)]R = [(<.w, v>.F<.u, t>.)]R) -> ([<.A, B>.]RG[<.C, D>.]R) = [(<.A, B>.F<.C, D>.)]R))
57	8, 37, 56	sylc 83	1 |- (((A e. S /\ B e. S) /\ (C e. S /\ D e. S)) -> ([<.A, B>.]RG[<.C, D>.]R) = [(<.A, B>.F<.C, D>.)]R)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  E.wex 1326  _Vcvv 2292  <.cop 3046   class class class wbr 3338   X. cxp 3984  dom cdm 3986  (class class class)co 4884  {copab2 4885  Er wer 5315  [cec 5316  /.cqs 5317

This theorem is referenced by:  oprec 5377

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fv 4014  df-opr 4886  df-oprab 4887  df-er 5318  df-ec 5320  df-qs 5323

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